On Hopf algebras of dimension p3p^3

We discuss some general results on finite-dimensional Hopf algebras over an algebraically closed field k of characteristic zero and then apply them to Hopf algebras H of dimension p^{3} over k. There are 10 cases according to the group-like elements of H and H^{*}. We show that in 8 of the 10 cases, it is possible to determine the structure of the Hopf algebra. We give also a partial classification of the quasitriangular Hopf algebras of dimension p^{3} over k, after studying extensions of a group algebra of order p by a Taft algebra of dimension p^{2}. In particular, we prove that every ribbon Hopf algebra of dimension p^{3} over k is either a group algebra or a Frobenius-Lusztig kernel. Finally, using some previous results on bounds for the dimension of the first term H_{1} in the coradical filtration of H, we give the complete classification of the quasitriangular Hopf algebras of dimension 27.Comment: 27 pages, minor changes. Accepted for publication in the Tsukuba Journal of Mathematic

Multiparameter quantum groups at roots of unity

We address the problem of studying multiparameter quamtum groups (=MpQG's) at roots of unity, namely quantum universal enveloping algebras $U_{\boldsymbol{\rm q}}(\mathfrak{g})$ depending on a matrix of parameters $\boldsymbol{\rm q} = {\big( q_{ij} \big)}_{i, j \in I} \,$. This is performed via the construction of quantum root vectors and suitable "integral forms" of $U_{\boldsymbol{\rm q}}(\mathfrak{g}) \,$, a restricted one - generated by quantum divided powers and quantum binomial coefficients - and an unrestricted one - where quantum root vectors are suitably renormalized. The specializations at roots of unity of either forms are the "MpQG's at roots of unity" we are investigating. In particular, we study special subalgebras and quotients of our MpQG's at roots of unity - namely, the multiparameter version of small quantum groups - and suitable associated quantum Frobenius morphisms, that link the (specializations of) MpQG's at roots of 1 with MpQG's at 1, the latter being classical Hopf algebras bearing a well precise Poisson-geometrical content. A key point in the discussion - often at the core of our strategy - is that every MpQG is actually a 2-cocycle deformation of the algebra structure of (a lift of) the "canonical" one-parameter quantum group by Jimbo-Lusztig, so that we can often rely on already established results available for the latter. On the other hand, depending on the chosen multiparameter $\boldsymbol{\rm q}$ our quantum groups yield (through the choice of integral forms and their specialization) different semiclassical structures, namely different Lie coalgebra structures and Poisson structures on the Lie algebra and algebraic group underlying the canonical one-parameter quantum group.Comment: 84 pages. New version slightly re-edited and streamlined: the content only is affected in Sec. 3.1, but page flushing occurs in the sequel as well (overall, the text is now one page shorter

Classifying Hopf algebras of a given dimension

Classifying all Hopf algebras of a given finite dimension over the complex numbers is a challenging problem which remains open even for many small dimensions, not least because few general approaches to the problem are known. Some useful techniques include counting the dimensions of spaces related to the coradical filtration, studying sub- and quotient Hopf algebras, especially those sub-Hopf algebras generated by a simple subcoalgebra, working with the antipode, and studying Hopf algebras in Yetter-Drinfeld categories to help to classify Radford biproducts. In this paper, we add to the classification tools in our previous work [arXiv:1108.6037v1] and apply our results to Hopf algebras of dimension rpq and 8p where p,q,r are distinct primes. At the end of this paper we summarize in a table the status of the classification for dimensions up to 100 to date.Comment: This version of the paper contains a correction on the published version. The statement and proof of Proposition 2.17 are changed and the proof of the results that follow from it are corrected accordingly. We thank H.-S. Ng for kindly communicating the gap to us and for the careful reading of our pape

Deformation by cocycles of pointed Hopf algebras over non-abelian groups

We introduce a method to construct explicitly multiplicative 2-cocycles for bosonizations of Nichols algebras B(V) over Hopf algebras H. These cocycles arise as liftings of H-invariant linear functionals on V tensor V and give a close formula to deform braided commutator-type relations. Using this construction, we show that all known finite dimensional pointed Hopf algebras over the dihedral groups D_m with m=4t > 11, over the symmetric group S_3 and some families over S_4 are cocycle deformations of bosonizations of Nichols algebras.Comment: 20 pages. This version: extended version following the referee's suggestions. Intended for non-expert